A Necessary and Sufficient Condition for Hardy's Operator in the Variable Lebesgue Space. (6th May 2014)

• Record Type: Journal Article
• Authors: Mamedov, Farman; Zeren, Yusuf
• Other Names: Kokilashvili Vakhtang M. Academic Editor.
• Is Part Of: Abstract and applied analysis. Volume 2014(2014)
• Publication Date: 2014-05-06
• Subjects: Mathematical analysis -- Periodicals; Mathematical analysis; Applied Mathematics; Mathematical Analysis; Periodicals; 515.05
• DOI: 10.1155/2014/342910
• Languages: English
• ISSNs: 1085-3375

Abstract

Abstract : The variable exponent Hardy inequality x β ( x ) - 1 ∫ 0 x ‍ f ( t ) d t L p ( . ) ( 0, l ) ≤ C x β ( x ) f L p ( . ) ( 0, l ), f ≥ 0 is proved assuming that the exponents p : ( 0, l ) → ( 1, ∞ ), β : ( 0, l ) → ℝ not rapidly oscilate near origin and 1 / p ′ ( 0 ) - β > 0 . The main result is a necessary and sufficient condition on p, β generalizing known results on this inequality.